Given f(x) = �ãx - 2 and , which of the following is the domain of the quotient function ?
a.(-infinity,2]
b.(-infinity,7) U (7,infinity)
c.[2,7) U (7, infinity)
d.(2, infinity)
The funny symbol next to the -2 represents a root sign.
To determine the domain of the quotient function, we need to consider the division of two functions. In this case, we have the quotient of f(x) divided by g(x), where f(x) = x - 2 and g(x) = x^2 - 49.
To find the domain of the quotient function, we first note that the only values not in the domain are the x-values that make the denominator equal to zero. In other words, we need to find the values of x that make g(x) = 0.
Considering the function g(x) = x^2 - 49, we can solve this equation for x by factoring it as a difference of squares:
x^2 - 49 = 0
(x - 7)(x + 7) = 0
This equation is satisfied when x = 7 or x = -7.
Therefore, the only values not in the domain of the quotient function are x = 7 and x = -7.
So, the domain of the quotient function is the set of all real numbers except x = 7 and x = -7.
Among the given answer choices:
a. (-infinity,2] does not exclude x = 7 and x = -7, so it is not the correct answer.
b. (-infinity,7) U (7,infinity) excludes both x = 7 and x = -7, so it is a possible correct answer.
c. [2,7) U (7, infinity) excludes x = -7, so it is not the correct answer.
d. (2, infinity) excludes x = -7, so it is not the correct answer.
Therefore, the correct answer is b. (-infinity,7) U (7,infinity).